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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>Usually, for one ODE, it is classified by its order. For instance, for the following ODE, </p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq1_4.html ./knowl/eq1_4.html ./knowl/eq1_4.html">
\begin{equation}
\frac{\textrm{d}^2y}{\textrm{d}x^2}+\frac{\textrm{d}y}{\textrm{d} x}+x=0,\tag{1.1.3}
\end{equation}
</div>
<p class="continuation">the highest order of derivative is <span class="process-math">\(2\text{,}\)</span> thus it is called second order ODE. Generally, an ODE may take the form</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq1_4.html ./knowl/eq1_4.html ./knowl/eq1_4.html">
\begin{equation*}
F(x, u(x), u^{\prime}(x), \cdots, u^{(n)}(x))=0.
\end{equation*}
</div>
<p class="continuation"> In (<a href="" class="xref" data-knowl="./knowl/eq1_4.html" title="Equation 1.1.3">(1.1.3)</a>), <span class="process-math">\(F(\cdots)\)</span> is a given function and the unknown is <span class="process-math">\(u(x)\text{.}\)</span> The highest order of derivatives is <span class="process-math">\(n\text{,}\)</span> therefore we say that the <span class="process-math">\(\textbf{order}\)</span> of this ODE is <span class="process-math">\(n\text{.}\)</span> The solution of (<a href="" class="xref" data-knowl="./knowl/eq1_4.html" title="Equation 1.1.3">(1.1.3)</a>) is the function <span class="process-math">\(\phi(x)\)</span> such that <span class="process-math">\(u=\phi(x)\)</span> satisfies (<a href="" class="xref" data-knowl="./knowl/eq1_4.html" title="Equation 1.1.3">(1.1.3)</a>), i.e.</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq1_4.html ./knowl/eq1_4.html ./knowl/eq1_4.html">
\begin{equation*}
F(x, \phi(x), \phi^{\prime}(x),\cdots,\phi^{(n)}(x)) \equiv 0.
\end{equation*}
</div>
<p class="continuation">For example, for the ODE <span class="process-math">\(\frac{dy }{dx}=x\text{,}\)</span> the solution is <span class="process-math">\(y=\frac{1}{2} x^2\text{.}\)</span></p>
<span class="incontext"><a href="sec_1-intro.html#p-3" class="internal">in-context</a></span>
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